How can I achieve the visual of objects pressing against membranes more beautifully?
While I was ultimately forced to use a realtime displacement shader in Unity to achieve the effect and my client later moved onto an off-the-shelf raymarching solution in Unity, I was able to slightly improve my workflow in Blender.

Using a Displacement Modifier to initially perturb the subdivided plane provided a gentle organic 'texture', I then followed this with a Warp modifier to move the vertices in the direction of my depth map (the Warp modifier obviated the need to worry about normals as it extends the vertices towards the 'to' empty).

While not perfect (still getting stretching artifacts, etc), it worked out pretty well within Blender and I was able to export Alembic bakes for playback in Unity.